Question

If the total number of observations is \(20\), \[ \sum x_i=1000 \] and \[ \sum x_i^2=84000, \] then the variance of the distribution is

• A. \(1500\)
• B. \(1600\)
• C. \(1700\)
• D. \(1800\)

Hint:

The shortcut formula for variance is \[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \] which avoids calculating each deviation individually.

Answer 1

The Correct Option is C

Step 1: Write the variance formula.
The variance for $ n $ observations is: \[ \sigma^2 = \frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2 \] This is the shortcut formula derived from the definition $ \sigma^2 = \frac{1}{n}\sum(x_i - \bar{x})^2 $.
Step 2: Identify the given values.
$ n = 20 $, $ \sum x_i = 1000 $, $ \sum x_i^2 = 84000 $. We will substitute directly into the formula.
Step 3: Compute $ \frac{\sum x_i^2}{n} $.
\[ \frac{\sum x_i^2}{n} = \frac{84000}{20} = 4200 \]
Step 4: Compute the squared mean.
\[ \bar{x} = \frac{\sum x_i}{n} = \frac{1000}{20} = 50, \quad \bar{x}^2 = 2500 \]
Step 5: Subtract to get variance.
\[ \sigma^2 = 4200 - 2500 = 1700 \]
Step 6: State the answer.
\[ \boxed{1700} \]